R/centrality.R

In igraph/rigraph: Network Analysis and Visualization

#' Find subgraph centrality scores of network positions
#'
#' @description
#' `r lifecycle::badge("deprecated")`
#'
#' `subgraph.centrality()` was renamed to [subgraph_centrality()] to create a more
#' consistent API.
#' @inheritParams subgraph_centrality
#' @keywords internal
#' @export
subgraph.centrality <- function(graph, diag = FALSE) {
  # nocov start
  lifecycle::deprecate_soft(
    "2.0.0",
    "subgraph.centrality()",
    "subgraph_centrality()"
  )
  subgraph_centrality(graph = graph, diag = diag)
} # nocov end

#' The Page Rank algorithm
#'
#' @description
#' `r lifecycle::badge("deprecated")`
#'
#' `page.rank()` was renamed to [page_rank()] to create a more
#' consistent API.
#' @inheritParams page_rank
#' @keywords internal
#' @export
page.rank <- function(
  graph,
  algo = c("prpack", "arpack"),
  vids = V(graph),
  directed = TRUE,
  damping = 0.85,
  personalized = NULL,
  weights = NULL,
  options = NULL
) {
  # nocov start
  lifecycle::deprecate_soft("2.0.0", "page.rank()", "page_rank()")
  page_rank(
    graph = graph,
    algo = algo,
    vids = vids,
    directed = directed,
    damping = damping,
    personalized = personalized,
    weights = weights,
    options = options
  )
} # nocov end

#' Kleinberg's hub and authority centrality scores.
#'
#' @description
#' `r lifecycle::badge("deprecated")`
#'
#' `hub.score()` was renamed to [hub_score()] to create a more
#' consistent API.
#' @inheritParams hub_score
#' @keywords internal
#' @export
hub.score <- function(
  graph,
  scale = TRUE,
  weights = NULL,
  options = arpack_defaults()
) {
  # nocov start
  lifecycle::deprecate_warn("2.0.0", "hub.score()", "hits_scores()")
  hub_score(graph = graph, scale = scale, weights = weights, options = options)
} # nocov end

#' Kleinberg's hub and authority centrality scores.
#'
#' @description
#' `r lifecycle::badge("deprecated")`
#'
#' `authority.score()` was renamed to [authority_score()] to create a more
#' consistent API.
#' @inheritParams authority_score
#' @keywords internal
#' @export
authority.score <- function(
  graph,
  scale = TRUE,
  weights = NULL,
  options = arpack_defaults()
) {
  # nocov start
  lifecycle::deprecate_warn("2.0.0", "authority.score()", "hits_scores()")
  authority_score(
    graph = graph,
    scale = scale,
    weights = weights,
    options = options
  )
} # nocov end

#' Strength or weighted vertex degree
#'
#' @description
#' `r lifecycle::badge("deprecated")`
#'
#' `graph.strength()` was renamed to [strength()] to create a more
#' consistent API.
#' @inheritParams strength
#' @keywords internal
#' @export
graph.strength <- function(
  graph,
  vids = V(graph),
  mode = c("all", "out", "in", "total"),
  loops = TRUE,
  weights = NULL
) {
  # nocov start
  lifecycle::deprecate_soft("2.0.0", "graph.strength()", "strength()")
  strength(
    graph = graph,
    vids = vids,
    mode = mode,
    loops = loops,
    weights = weights
  )
} # nocov end

#' Eigenvalues and eigenvectors of the adjacency matrix of a graph
#'
#' @description
#' `r lifecycle::badge("deprecated")`
#'
#' `graph.eigen()` was renamed to [spectrum()] to create a more
#' consistent API.
#' @inheritParams spectrum
#' @keywords internal
#' @export
graph.eigen <- function(
  graph,
  algorithm = c(
    "arpack",
    "auto",
    "lapack",
    "comp_auto",
    "comp_lapack",
    "comp_arpack"
  ),
  which = list(),
  options = arpack_defaults()
) {
  # nocov start
  lifecycle::deprecate_soft("2.0.0", "graph.eigen()", "spectrum()")
  spectrum(
    graph = graph,
    algorithm = algorithm,
    which = which,
    options = options
  )
} # nocov end

#' Graph diversity
#'
#' @description
#' `r lifecycle::badge("deprecated")`
#'
#' `graph.diversity()` was renamed to [diversity()] to create a more
#' consistent API.
#' @inheritParams diversity
#' @keywords internal
#' @export
graph.diversity <- function(graph, weights = NULL, vids = V(graph)) {
  # nocov start
  lifecycle::deprecate_soft("2.0.0", "graph.diversity()", "diversity()")
  diversity(graph = graph, weights = weights, vids = vids)
} # nocov end

#' Find Eigenvector Centrality Scores of Network Positions
#'
#' @description
#' `r lifecycle::badge("deprecated")`
#'
#' `evcent()` was renamed to [eigen_centrality()] to create a more
#' consistent API.
#' @inheritParams eigen_centrality
#' @keywords internal
#' @export
evcent <- function(
  graph,
  directed = FALSE,
  scale = TRUE,
  weights = NULL,
  options = arpack_defaults()
) {
  # nocov start
  lifecycle::deprecate_soft("2.0.0", "evcent()", "eigen_centrality()")
  eigen_centrality(
    graph = graph,
    directed = directed,
    scale = scale,
    weights = weights,
    options = options
  )
} # nocov end

#' Vertex and edge betweenness centrality
#'
#' @description
#' `r lifecycle::badge("deprecated")`
#'
#' `edge.betweenness()` was renamed to [edge_betweenness()] to create a more
#' consistent API.
#' @inheritParams edge_betweenness
#' @keywords internal
#' @export
edge.betweenness <- function(
  graph,
  e = E(graph),
  directed = TRUE,
  weights = NULL,
  cutoff = -1
) {
  # nocov start
  lifecycle::deprecate_soft("2.0.0", "edge.betweenness()", "edge_betweenness()")
  edge_betweenness(
    graph = graph,
    e = e,
    directed = directed,
    weights = weights,
    cutoff = cutoff
  )
} # nocov end

#' Find Bonacich Power Centrality Scores of Network Positions
#'
#' @description
#' `r lifecycle::badge("deprecated")`
#'
#' `bonpow()` was renamed to [power_centrality()] to create a more
#' consistent API.
#' @inheritParams power_centrality
#' @keywords internal
#' @export
bonpow <- function(
  graph,
  nodes = V(graph),
  loops = FALSE,
  exponent = 1,
  rescale = FALSE,
  tol = 1e-7,
  sparse = TRUE
) {
  # nocov start
  lifecycle::deprecate_soft("2.0.0", "bonpow()", "power_centrality()")
  power_centrality(
    graph = graph,
    nodes = nodes,
    loops = loops,
    exponent = exponent,
    rescale = rescale,
    tol = tol,
    sparse = sparse
  )
} # nocov end

#' Find Bonacich alpha centrality scores of network positions
#'
#' @description
#' `r lifecycle::badge("deprecated")`
#'
#' `alpha.centrality()` was renamed to [alpha_centrality()] to create a more
#' consistent API.
#' @inheritParams alpha_centrality
#' @keywords internal
#' @export
alpha.centrality <- function(
  graph,
  nodes = V(graph),
  alpha = 1,
  loops = FALSE,
  exo = 1,
  weights = NULL,
  tol = 1e-7,
  sparse = TRUE
) {
  # nocov start
  lifecycle::deprecate_soft("2.0.0", "alpha.centrality()", "alpha_centrality()")
  alpha_centrality(
    graph = graph,
    nodes = nodes,
    alpha = alpha,
    loops = loops,
    exo = exo,
    weights = weights,
    tol = tol,
    sparse = sparse
  )
} # nocov end
#   IGraph R package
#   Copyright (C) 2005-2012  Gabor Csardi <csardi.gabor@gmail.com>
#   334 Harvard street, Cambridge, MA 02139 USA
#
#   This program is free software; you can redistribute it and/or modify
#   it under the terms of the GNU General Public License as published by
#   the Free Software Foundation; either version 2 of the License, or
#   (at your option) any later version.
#
#   This program is distributed in the hope that it will be useful,
#   but WITHOUT ANY WARRANTY; without even the implied warranty of
#   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
#   GNU General Public License for more details.
#
#   You should have received a copy of the GNU General Public License
#   along with this program; if not, write to the Free Software
#   Foundation, Inc.,  51 Franklin Street, Fifth Floor, Boston, MA
#   02110-1301 USA
#
###################################################################

#' Deprecated version of `betweenness()`
#'
#' @description
#' `r lifecycle::badge("deprecated")`
#'
#' Use [betweenness()] with the `cutoff` argument instead.
#' @param vids The vertices for which the vertex betweenness estimation will be
#'   calculated.
#' @inheritParams betweenness
#' @keywords internal
#' @export
estimate_betweenness <- function(
  graph,
  vids = V(graph),
  directed = TRUE,
  cutoff,
  weights = NULL
) {
  lifecycle::deprecate_soft(
    "1.6.0",
    "estimate_betweenness()",
    "betweenness()",
    details = "with the cutoff argument."
  )

  betweenness(
    graph,
    v = vids,
    directed = directed,
    cutoff = cutoff,
    weights = weights
  )
}

#' @export
betweenness.estimate <- estimate_betweenness


#' Vertex and edge betweenness centrality
#'
#' The vertex and edge betweenness are (roughly) defined by the number of
#' geodesics (shortest paths) going through a vertex or an edge.
#'
#' The vertex betweenness of vertex `v` is defined by
#'
#' \deqn{\sum_{i\ne j, i\ne v, j\ne v} g_{ivj}/g_{ij}}{sum( g_ivj / g_ij,
#' i!=j,i!=v,j!=v)}
#'
#' The edge betweenness of edge `e` is defined by
#'
#' \deqn{\sum_{i\ne j} g_{iej}/g_{ij}.}{sum( g_iej / g_ij, i!=j).}
#'
#' `betweenness()` calculates vertex betweenness, `edge_betweenness()`
#' calculates edge betweenness.
#'
#' Here \eqn{g_{ij}}{g_ij} is the total number of shortest paths between vertices
#' \eqn{i} and \eqn{j} while \eqn{g_{ivj}} is the number of those shortest paths
#' which pass though vertex \eqn{v}.
#'
#' Both functions allow you to consider only paths of length `cutoff` or
#' smaller; this can be run for larger graphs, as the running time is not
#' quadratic (if `cutoff` is small). If `cutoff` is negative (the default),
#' then the function calculates the exact betweenness scores. Since igraph 1.6.0,
#' a `cutoff` value of zero is treated literally, i.e. paths of length larger
#' than zero are ignored.
#'
#' For calculating the betweenness a similar algorithm to the one proposed by
#' Brandes (see References) is used.
#'
#' @aliases betweenness.estimate
#' @aliases edge.betweenness.estimate
#' @param graph The graph to analyze.
#' @param v The vertices for which the vertex betweenness will be calculated.
#' @param directed Logical, whether directed paths should be considered while
#'   determining the shortest paths.
#' @param weights Optional positive weight vector for calculating weighted
#'   betweenness. If the graph has a `weight` edge attribute, then this is
#'   used by default. Weights are used to calculate weighted shortest paths,
#'   so they are interpreted as distances.
#' @param normalized Logical scalar, whether to normalize the betweenness
#'   scores. If `TRUE`, then the results are normalized by the number of ordered
#'   or unordered vertex pairs in directed and undirected graphs, respectively.
#'   In an undirected graph,
#'   \deqn{B^n=\frac{2B}{(n-1)(n-2)},}{Bnorm=2 B / ((n-1)(n-2)),}
#'   where
#'   \eqn{B^n}{Bnorm} is the normalized, \eqn{B} the raw betweenness, and
#'   \eqn{n} is the number of vertices in the graph. Note that the same
#'   normalization factor is used even when setting a `cutoff` on the considered
#'   shortest path lengths, even though the number of vertex pairs reachable
#'   from each other may be less than \eqn{(n-1)(n-2)/2}.
#' @param cutoff The maximum shortest path length to consider when calculating
#'   betweenness. If negative, then there is no such limit.
#' @return A numeric vector with the betweenness score for each vertex in
#'   `v` for `betweenness()`.
#'
#'   A numeric vector with the edge betweenness score for each edge in `e`
#'   for `edge_betweenness()`.
#'
#' @note `edge_betweenness()` might give false values for graphs with
#' multiple edges.
#' @author Gabor Csardi \email{csardi.gabor@@gmail.com}
#' @seealso [closeness()], [degree()], [harmonic_centrality()]
#' @references Freeman, L.C. (1979). Centrality in Social Networks I:
#' Conceptual Clarification. *Social Networks*, 1, 215-239.
#' \doi{10.1016/0378-8733(78)90021-7}
#'
#' Ulrik Brandes, A Faster Algorithm for Betweenness Centrality. *Journal
#' of Mathematical Sociology* 25(2):163-177, 2001.
#' \doi{10.1080/0022250X.2001.9990249}
#' @family centrality
#' @export
#' @keywords graphs
#' @examples
#'
#' g <- sample_gnp(10, 3 / 10)
#' betweenness(g)
#' edge_betweenness(g)
#'
betweenness <- function(
  graph,
  v = V(graph),
  directed = TRUE,
  weights = NULL,
  normalized = FALSE,
  cutoff = -1
) {
  ensure_igraph(graph)

  v <- as_igraph_vs(graph, v)
  directed <- as.logical(directed)
  if (is.null(weights) && "weight" %in% edge_attr_names(graph)) {
    weights <- E(graph)$weight
  }
  if (!is.null(weights) && any(!is.na(weights))) {
    weights <- as.numeric(weights)
  } else {
    weights <- NULL
  }
  cutoff <- as.numeric(cutoff)
  on.exit(.Call(R_igraph_finalizer))
  res <- .Call(
    R_igraph_betweenness_cutoff,
    graph,
    v - 1,
    directed,
    weights,
    cutoff
  )
  if (normalized) {
    vc <- as.numeric(vcount(graph))
    if (is_directed(graph) && directed) {
      res <- res / (vc * vc - 3 * vc + 2)
    } else {
      res <- 2 * res / (vc * vc - 3 * vc + 2)
    }
  }
  if (igraph_opt("add.vertex.names") && is_named(graph)) {
    names(res) <- V(graph)$name[v]
  }
  res
}

#' @rdname betweenness
#' @param e The edges for which the edge betweenness will be calculated.
#' @export
edge_betweenness <- function(
  graph,
  e = E(graph),
  directed = TRUE,
  weights = NULL,
  cutoff = -1
) {
  # Argument checks
  ensure_igraph(graph)

  e <- as_igraph_es(graph, e)
  directed <- as.logical(directed)
  if (is.null(weights) && "weight" %in% edge_attr_names(graph)) {
    weights <- E(graph)$weight
  }
  if (!is.null(weights) && any(!is.na(weights))) {
    weights <- as.numeric(weights)
  } else {
    weights <- NULL
  }
  cutoff <- as.numeric(cutoff)

  on.exit(.Call(R_igraph_finalizer))
  # Function call
  res <- .Call(
    R_igraph_edge_betweenness_cutoff,
    graph,
    directed,
    weights,
    cutoff
  )
  res[as.numeric(e)]
}

#' Deprecated version of `edge_betweenness()`
#'
#' @description
#' `r lifecycle::badge("deprecated")`
#'
#' Use [edge_betweenness()] with the `cutoff` argument instead.
#' @inheritParams edge_betweenness
#' @keywords internal
#' @export
estimate_edge_betweenness <- function(
  graph,
  e = E(graph),
  directed = TRUE,
  cutoff,
  weights = NULL
) {
  lifecycle::deprecate_soft(
    "1.6.0",
    "estimate_edge_betweenness()",
    "edge_betweenness()",
    details = "with the cutoff argument."
  )
  edge_betweenness(
    graph,
    e,
    directed = directed,
    cutoff = cutoff,
    weights = weights
  )
}

#' @export
edge.betweenness.estimate <- estimate_edge_betweenness

#' Closeness centrality of vertices
#'
#' Closeness centrality measures how many steps are required to access every other
#' vertex from a given vertex.
#'
#' The closeness centrality of a vertex is defined as the inverse of the
#' sum of distances to all the other vertices in the graph:
#'
#' \deqn{\frac{1}{\sum_{i\ne v} d_{vi}}}{1/sum( d(v,i), i != v)}
#'
#' If there is no (directed) path between vertex `v` and `i`, then
#' `i` is omitted from the calculation. If no other vertices are reachable
#' from `v`, then its closeness is returned as NaN.
#'
# " You may use the \code{cutoff} argument to consider only paths of length
#' `cutoff` or smaller. This can be run for larger graphs, as the running
#' time is not quadratic (if `cutoff` is small). If `cutoff` is
#' negative (which is the default), then the function calculates the exact
#' closeness scores. Since igraph 1.6.0, a `cutoff` value of zero is treated
#' literally, i.e. path with a length greater than zero are ignored.
#'
#' Closeness centrality is meaningful only for connected graphs. In disconnected
#' graphs, consider using the harmonic centrality with
#' [harmonic_centrality()]
#'
#' @aliases closeness.estimate
#' @param graph The graph to analyze.
#' @param vids The vertices for which closeness will be calculated.
#' @param mode Character string, defined the types of the paths used for
#'   measuring the distance in directed graphs. \dQuote{in} measures the paths
#'   *to* a vertex, \dQuote{out} measures paths *from* a vertex,
#'   *all* uses undirected paths. This argument is ignored for undirected
#'   graphs.
#' @param normalized Logical scalar, whether to calculate the normalized
#'   closeness, i.e. the inverse average distance to all reachable vertices.
#'   The non-normalized closeness is the inverse of the sum of distances to
#'   all reachable vertices.
#' @param weights Optional positive weight vector for calculating weighted
#'   closeness. If the graph has a `weight` edge attribute, then this is
#'   used by default. Weights are used for calculating weighted shortest
#'   paths, so they are interpreted as distances.
#' @param cutoff The maximum path length to consider when calculating the
#'   closeness. If zero or negative then there is no such limit.
#' @return Numeric vector with the closeness values of all the vertices in
#'   `v`.
#' @author Gabor Csardi \email{csardi.gabor@@gmail.com}
#' @references Freeman, L.C. (1979). Centrality in Social Networks I:
#' Conceptual Clarification. *Social Networks*, 1, 215-239.
#' @family centrality
#' @export
#' @keywords graphs
#' @examples
#'
#' g <- make_ring(10)
#' g2 <- make_star(10)
#' closeness(g)
#' closeness(g2, mode = "in")
#' closeness(g2, mode = "out")
#' closeness(g2, mode = "all")
#'
closeness <- function(
  graph,
  vids = V(graph),
  mode = c("out", "in", "all", "total"),
  weights = NULL,
  normalized = FALSE,
  cutoff = -1
) {
  # Argument checks
  ensure_igraph(graph)

  vids <- as_igraph_vs(graph, vids)
  mode <- switch(
    igraph.match.arg(mode),
    "out" = 1,
    "in" = 2,
    "all" = 3,
    "total" = 3
  )
  if (is.null(weights) && "weight" %in% edge_attr_names(graph)) {
    weights <- E(graph)$weight
  }
  if (!is.null(weights) && any(!is.na(weights))) {
    weights <- as.numeric(weights)
  } else {
    weights <- NULL
  }
  normalized <- as.logical(normalized)
  cutoff <- as.numeric(cutoff)

  on.exit(.Call(R_igraph_finalizer))
  # Function call
  res <- .Call(
    R_igraph_closeness_cutoff,
    graph,
    vids - 1,
    mode,
    weights,
    normalized,
    cutoff
  )$res
  if (igraph_opt("add.vertex.names") && is_named(graph)) {
    names(res) <- V(graph)$name[vids]
  }
  res
}

#' Deprecated version of `closeness()`
#'
#' @description
#' `r lifecycle::badge("deprecated")`
#'
#' Use [closeness()] with the `cutoff` argument instead.
#' @inheritParams closeness
#' @keywords internal
#' @export
estimate_closeness <- function(
  graph,
  vids = V(graph),
  mode = c("out", "in", "all", "total"),
  cutoff,
  weights = NULL,
  normalized = FALSE
) {
  lifecycle::deprecate_soft(
    "1.6.0",
    "estimate_closeness()",
    "closeness()",
    details = "with the cutoff argument."
  )

  closeness(
    graph,
    vids,
    mode = mode,
    weights = weights,
    normalized = normalized,
    cutoff = cutoff
  )
}

#' @export
closeness.estimate <- estimate_closeness
#' @rdname arpack
#' @family arpack
#' @export
arpack_defaults <- function() {
  list(
    bmat = "I",
    n = 0,
    which = "XX",
    nev = 1,
    tol = 0.0,
    ncv = 3,
    ldv = 0,
    ishift = 1,
    maxiter = 3000,
    nb = 1,
    mode = 1,
    start = 0,
    sigma = 0.0,
    sigmai = 0.0
  )
}

#' ARPACK eigenvector calculation
#'
#' Interface to the ARPACK library for calculating eigenvectors of sparse
#' matrices
#'
#' @details
#' ARPACK is a library for solving large scale eigenvalue problems.  The
#' package is designed to compute a few eigenvalues and corresponding
#' eigenvectors of a general \eqn{n} by \eqn{n} matrix \eqn{A}. It is most
#' appropriate for large sparse or structured matrices \eqn{A} where structured
#' means that a matrix-vector product `w <- Av` requires order \eqn{n}
#' rather than the usual order \eqn{n^2} floating point operations.
#'
#' This function is an interface to ARPACK. igraph does not contain all ARPACK
#' routines, only the ones dealing with symmetric and non-symmetric eigenvalue
#' problems using double precision real numbers.
#'
#' The eigenvalue calculation in ARPACK (in the simplest case) involves the
#' calculation of the \eqn{Av} product where \eqn{A} is the matrix we work with
#' and \eqn{v} is an arbitrary vector. The function supplied in the `fun`
#' argument is expected to perform this product. If the product can be done
#' efficiently, e.g. if the matrix is sparse, then `arpack()` is usually
#' able to calculate the eigenvalues very quickly.
#'
#' @details
#' The `options` argument specifies what kind of calculation to perform.
#' It is a list with the following members, they correspond directly to ARPACK
#' parameters. On input it has the following fields:
#'   \describe{
#'     \item{bmat}{
#'       Character constant, possible values:
#'       \sQuote{`I`}, standard eigenvalue problem, \eqn{Ax=\lambda x}{A*x=lambda*x}; and
#'       \sQuote{`G`}, generalized eigenvalue problem, \eqn{Ax=\lambda B x}{A*x=lambda B*x}.
#'       Currently only \sQuote{`I`} is supported.
#'     }
#'     \item{n}{
#'       Numeric scalar. The dimension of the eigenproblem.
#'       You only need to set this if you call [arpack()] directly.
#'       (I.e. not needed for [eigen_centrality()], [page_rank()], etc.)
#'     }
#'     \item{which}{
#'       Specify which eigenvalues/vectors to compute,
#'       character constant with exactly two characters.
#'       Possible values for symmetric input matrices:
#'         \describe{
#'           \item{"LA"}{
#'             Compute `nev` largest (algebraic) eigenvalues.
#'           }
#'           \item{"SA"}{
#'             Compute `nev` smallest (algebraic) eigenvalues.
#'           }
#'           \item{"LM"}{
#'             Compute `nev` largest (in magnitude) eigenvalues.
#'           }
#'           \item{"SM"}{
#'             Compute `nev` smallest (in magnitude) eigenvalues.
#'           }
#'           \item{"BE"}{
#'             Compute `nev` eigenvalues, half from each end of the spectrum.
#'             When `nev` is odd, compute one more from the high end than from the low end.
#'           }
#'         }
#'       Possible values for non-symmetric input matrices:
#'         \describe{
#'           \item{"LM"}{
#'             Compute `nev` eigenvalues of largest magnitude.
#'           }
#'           \item{"SM"}{
#'             Compute `nev` eigenvalues of smallest magnitude.
#'           }
#'           \item{"LR"}{
#'             Compute `nev` eigenvalues of largest real part.
#'           }
#'           \item{"SR"}{
#'             Compute `nev` eigenvalues of smallest real part.
#'           }
#'           \item{"LI"}{
#'             Compute `nev` eigenvalues of largest imaginary part.
#'           }
#'           \item{"SI"}{
#'             Compute `nev` eigenvalues of smallest imaginary part.
#'           }
#'         }
#'       This parameter is sometimes overwritten by the various functions,
#'       e.g. [page_rank()] always sets \sQuote{`LM`}.
#'     }
#'     \item{nev}{
#'       Numeric scalar. The number of eigenvalues to be computed.
#'     }
#'     \item{tol}{
#'       Numeric scalar. Stopping criterion:
#'       the relative accuracy of the Ritz value is considered acceptable
#'       if its error is less than `tol` times its estimated value.
#'       If this is set to zero then machine precision is used.
#'     }
#'     \item{ncv}{
#'       Number of Lanczos vectors to be generated.
#'     }
#'     \item{ldv}{
#'       Numberic scalar. It should be set to zero in the current implementation.
#'     }
#'     \item{ishift}{
#'       Either zero or one.
#'       If zero then the shifts are provided by the user via reverse communication.
#'       If one then exact shifts with respect to the reduced tridiagonal matrix \eqn{T}.
#'       Please always set this to one.
#'     }
#'     \item{maxiter}{
#'       Maximum number of Arnoldi update iterations allowed.
#'     }
#'     \item{nb}{
#'       Blocksize to be used in the recurrence. Please always leave this on the default value, one.
#'     }
#'     \item{mode}{
#'       The type of the eigenproblem to be solved.  Possible values if the input matrix is symmetric:
#'         \describe{
#'           \item{1}{
#'             \eqn{Ax=\lambda x}{A*x=lambda*x}, \eqn{A} is symmetric.
#'           }
#'           \item{2}{
#'             \eqn{Ax=\lambda Mx}{A*x=lambda*M*x}, \eqn{A} is symmetric, \eqn{M} is symmetric positive definite.
#'           }
#'           \item{3}{
#'             \eqn{Kx=\lambda Mx}{K*x=lambda*M*x}, \eqn{K} is symmetric, \eqn{M} is symmetric positive semi-definite.
#'           }
#'           \item{4}{
#'             \eqn{Kx=\lambda KGx}{K*x=lambda*KG*x}, \eqn{K} is symmetric positive semi-definite, \eqn{KG} is symmetric indefinite.
#'           }
#'           \item{5}{
#'             \eqn{Ax=\lambda Mx}{A*x=lambda*M*x}, \eqn{A} is symmetric, \eqn{M} is symmetric positive semi-definite. (Cayley transformed mode.)
#'           }
#'         }
#'       Please note that only `mode==1` was tested and other values might not work properly.
#'       Possible values if the input matrix is not symmetric:
#'         \describe{
#'           \item{1}{
#'             \eqn{Ax=\lambda x}{A*x=lambda*x}.
#'           }
#'           \item{2}{
#'             \eqn{Ax=\lambda Mx}{A*x=lambda*M*x}, \eqn{M} is symmetric positive definite.
#'           }
#'           \item{3}{
#'             \eqn{Ax=\lambda Mx}{A*x=lambda*M*x}, \eqn{M} is symmetric semi-definite.
#'           }
#'           \item{4}{
#'             \eqn{Ax=\lambda Mx}{A*x=lambda*M*x}, \eqn{M} is symmetric semi-definite.
#'           }
#'         }
#'         Please note that only `mode==1` was tested and other values might not work properly.
#'     }
#'     \item{start}{
#'       Not used currently. Later it be used to set a starting vector.
#'     }
#'     \item{sigma}{
#'       Not used currently.
#'     }
#'     \item{sigmai}{
#'       Not use currently.
#'     }
#'   }
#'
#' @details:
#'   On output the following additional fields are added:
#'
#'   \describe{
#'     \item{info}{
#'       Error flag of ARPACK. Possible values:
#'         \describe{
#'           \item{0}{
#'             Normal exit.
#'           }
#'           \item{1}{
#'             Maximum number of iterations taken.
#'           }
#'           \item{3}{
#'             No shifts could be applied during a cycle
#'             of the implicitly restarted Arnoldi iteration.
#'             One possibility is to increase the size of `ncv` relative to `nev`.
#'           }
#'         }
#'       ARPACK can return more error conditions than these,
#'       but they are converted to regular igraph errors.
#'     }
#'     \item{iter}{
#'       Number of Arnoldi iterations taken.
#'     }
#'     \item{nconv}{
#'       Number of \dQuote{converged} Ritz values.
#'       This represents the number of Ritz values that satisfy the convergence critetion.
#'     }
#'     \item{numop}{
#'       Total number of matrix-vector multiplications.
#'     }
#'     \item{numopb}{
#'       Not used currently.
#'     }
#'     \item{numreo}{
#'       Total number of steps of re-orthogonalization.
#'     }
#'   }
#'
#' Please see the ARPACK documentation for additional details.
#'
#' @aliases arpack arpack-options arpack.unpack.complex
#' @aliases arpack_defaults
#' @param func The function to perform the matrix-vector multiplication. ARPACK
#'   requires to perform these by the user. The function gets the vector \eqn{x}
#'   as the first argument, and it should return \eqn{Ax}, where \eqn{A} is the
#'   \dQuote{input matrix}. (The input matrix is never given explicitly.) The
#'   second argument is `extra`.
#' @param extra Extra argument to supply to `func`.
#' @param sym Logical scalar, whether the input matrix is symmetric. Always
#'   supply `TRUE` here if it is, since it can speed up the computation.
#' @param options Options to ARPACK, a named list to overwrite some of the
#'   default option values. See details below.
#' @param env The environment in which `func` will be evaluated.
#' @param complex Whether to convert the eigenvectors returned by ARPACK into R
#'   complex vectors. By default this is not done for symmetric problems (these
#'   only have real eigenvectors/values), but only non-symmetric ones. If you
#'   have a non-symmetric problem, but you're sure that the results will be real,
#'   then supply `FALSE` here.
#' @return A named list with the following members:
#'   \describe{
#'     \item{values}{
#'       Numeric vector, the desired eigenvalues.
#'     }
#'     \item{vectors}{
#'       Numeric matrix, the desired eigenvectors as columns.
#'       If `complex=TRUE` (the default for non-symmetric problems), then the matrix is complex.
#'     }
#'     \item{options}{
#'       A named list with the supplied `options`
#'       and some information about the performed calculation,
#'       including an ARPACK exit code.
#'       See the details above.
#'     }
#'   }
#' @author Rich Lehoucq, Kristi Maschhoff, Danny Sorensen, Chao Yang for
#' ARPACK, Gabor Csardi \email{csardi.gabor@@gmail.com} for the R interface.
#' @seealso [eigen_centrality()], [page_rank()],
#' [hub_score()], [cluster_leading_eigen()] are some of the
#' functions in igraph that use ARPACK.
#' @references D.C. Sorensen, Implicit Application of Polynomial Filters in a
#' k-Step Arnoldi Method. *SIAM J. Matr. Anal. Apps.*, 13 (1992), pp
#' 357-385.
#'
#' R.B. Lehoucq, Analysis and Implementation of an Implicitly Restarted Arnoldi
#' Iteration. *Rice University Technical Report* TR95-13, Department of
#' Computational and Applied Mathematics.
#'
#' B.N. Parlett & Y. Saad, Complex Shift and Invert Strategies for Real
#' Matrices. *Linear Algebra and its Applications*, vol 88/89, pp 575-595,
#' (1987).
#' @keywords graphs
#' @examples
#'
#' # Identity matrix
#' f <- function(x, extra = NULL) x
#' arpack(f, options = list(n = 10, nev = 2, ncv = 4), sym = TRUE)
#'
#' # Graph laplacian of a star graph (undirected), n>=2
#' # Note that this is a linear operation
#' f <- function(x, extra = NULL) {
#'   y <- x
#'   y[1] <- (length(x) - 1) * x[1] - sum(x[-1])
#'   for (i in 2:length(x)) {
#'     y[i] <- x[i] - x[1]
#'   }
#'   y
#' }
#'
#' arpack(f, options = list(n = 10, nev = 1, ncv = 3), sym = TRUE)
#'
#' # double check
#' eigen(laplacian_matrix(make_star(10, mode = "undirected")))
#'
#' ## First three eigenvalues of the adjacency matrix of a graph
#' ## We need the 'Matrix' package for this
#' @examplesIf rlang::is_installed("Matrix")
#' library("Matrix")
#' set.seed(42)
#' g <- sample_gnp(1000, 5 / 1000)
#' M <- as_adjacency_matrix(g, sparse = TRUE)
#' f2 <- function(x, extra = NULL) {
#'   cat(".")
#'   as.vector(M %*% x)
#' }
#' baev <- arpack(
#'   f2,
#'   sym = TRUE,
#'   options = list(
#'     n = vcount(g),
#'     nev = 3,
#'     ncv = 8,
#'     which = "LM",
#'     maxiter = 2000
#'   )
#' )
#' @family arpack
#' @export
arpack <- function(
  func,
  extra = NULL,
  sym = FALSE,
  options = arpack_defaults(),
  env = parent.frame(),
  complex = !sym
) {
  if (is.function(options)) {
    lifecycle::deprecate_soft(
      "1.6.0",
      "arpack(options = 'must be a list')",
      details = c(
        "`arpack_defaults()` is now a function, use `options = arpack_defaults()` instead of `options = arpack_defaults`."
      )
    )
    options <- options()
  }

  if (
    !is.list(options) ||
      (is.null(names(options)) && length(options) != 0)
  ) {
    stop("options must be a named list")
  }
  if (any(names(options) == "")) {
    stop("all options must be named")
  }

  defaults <- arpack_defaults()
  if (any(!names(options) %in% names(defaults))) {
    stop(
      "unkown ARPACK option(s): ",
      paste(setdiff(names(options), names(defaults)), collapse = ", ")
    )
  }

  options <- modify_list(defaults, options)

  if (sym && complex) {
    complex <- FALSE
    cli::cli_warn("Symmetric matrix, setting {.arg complex} to {.code FALSE}.")
  }

  on.exit(.Call(R_igraph_finalizer))
  res <- .Call(R_igraph_arpack, func, extra, options, env, sym)

  if (complex) {
    rew <- arpack.unpack.complex(
      res$vectors,
      res$values,
      min(res$options$nev, res$options$nconv)
    )
    res$vectors <- rew$vectors
    res$values <- rew$values

    res$values <- apply(res$values, 1, function(x) x[1] + x[2] * 1i)
    dim(res$vectors) <- c(nrow(res$vectors) * 2, ncol(res$vectors) / 2)
    res$vectors <- apply(res$vectors, 2, function(x) {
      l <- length(x) / 2
      x[1:l] + x[(l + 1):length(x)] * 1i
    })
  } else {
    if (is.matrix(res$values)) {
      if (!all(res$values[, 2] == 0)) {
        cli::cli_warn("Dropping imaginary parts of eigenvalues.")
      }
      res$values <- res$values[, 1]
    }
    res$vectors <- res$vectors[, 1:length(res$values)]
  }

  res
}

arpack.unpack.complex <- function(vectors, values, nev) {
  # Argument checks
  vectors[] <- as.numeric(vectors)
  values[] <- as.numeric(values)
  nev <- as.numeric(nev)

  on.exit(.Call(R_igraph_finalizer))
  # Function call
  res <- .Call(R_igraph_arpack_unpack_complex, vectors, values, nev)

  res
}


#' Find subgraph centrality scores of network positions
#'
#' Subgraph centrality of a vertex measures the number of subgraphs a vertex
#' participates in, weighting them according to their size.
#'
#' The subgraph centrality of a vertex is defined as the number of closed walks
#' originating at the vertex, where longer walks are downweighted by the
#' factorial of their length.
#'
#' Currently the calculation is performed by explicitly calculating all
#' eigenvalues and eigenvectors of the adjacency matrix of the graph. This
#' effectively means that the measure can only be calculated for small graphs.
#'
#' @param graph The input graph. It will be treated as undirected.
#' @param diag Boolean scalar, whether to include the diagonal of the adjacency
#'   matrix in the analysis. Giving `FALSE` here effectively eliminates the
#'   loops edges from the graph before the calculation.
#' @return A numeric vector, the subgraph centrality scores of the vertices.
#' @author Gabor Csardi \email{csardi.gabor@@gmail.com} based on the Matlab
#' code by Ernesto Estrada
#' @seealso [eigen_centrality()], [page_rank()]
#' @references Ernesto Estrada, Juan A. Rodriguez-Velazquez: Subgraph
#' centrality in Complex Networks. *Physical Review E* 71, 056103 (2005).
#' @family centrality
#' @export
#' @keywords graphs
#' @examples
#'
#' g <- sample_pa(100, m = 4, dir = FALSE)
#' sc <- subgraph_centrality(g)
#' cor(degree(g), sc)
#'
subgraph_centrality <- function(graph, diag = FALSE) {
  A <- as_adjacency_matrix(graph)
  if (!diag) {
    diag(A) <- 0
  }
  # Ignore edge directions in directed graphs
  if (is_directed(graph)) {
    A <- A + Matrix::t(A)
  }
  # This calls lapack and creates a dense matrix, but accepts the sparse matrix A
  # We can choose to convert A to a dense matrix right away, but it doesn't matter
  eig <- eigen(A, symmetric = TRUE)
  res <- as.vector(eig$vectors^2 %*% exp(eig$values))
  if (igraph_opt("add.vertex.names") && is_named(graph)) {
    names(res) <- vertex_attr(graph, "name")
  }
  res
}


#' Eigenvalues and eigenvectors of the adjacency matrix of a graph
#'
#' Calculate selected eigenvalues and eigenvectors of a (supposedly sparse)
#' graph.
#'
#' The `which` argument is a list and it specifies which eigenvalues and
#' corresponding eigenvectors to calculate: There are eight options:
#' \enumerate{ \item Eigenvalues with the largest magnitude. Set `pos` to
#' `LM`, and `howmany` to the number of eigenvalues you want.  \item
#' Eigenvalues with the smallest magnitude. Set `pos` to `SM` and
#' `howmany` to the number of eigenvalues you want.  \item Largest
#' eigenvalues. Set `pos` to `LA` and `howmany` to the number of
#' eigenvalues you want.  \item Smallest eigenvalues. Set `pos` to
#' `SA` and `howmany` to the number of eigenvalues you want.  \item
#' Eigenvalues from both ends of the spectrum. Set `pos` to `BE` and
#' `howmany` to the number of eigenvalues you want. If `howmany` is
#' odd, then one more eigenvalue is returned from the larger end.  \item
#' Selected eigenvalues. This is not (yet) implemented currently.  \item
#' Eigenvalues in an interval. This is not (yet) implemented.  \item All
#' eigenvalues. This is not implemented yet. The standard `eigen` function
#' does a better job at this, anyway.  }
#'
#' Note that ARPACK might be unstable for graphs with multiple components, e.g.
#' graphs with isolate vertices.
#'
#' @aliases spectrum igraph.eigen.default
#' @param graph The input graph, can be directed or undirected.
#' @param algorithm The algorithm to use. Currently only `arpack` is
#'   implemented, which uses the ARPACK solver. See also [arpack()].
#' @param which A list to specify which eigenvalues and eigenvectors to
#'   calculate. By default the leading (i.e. largest magnitude) eigenvalue and
#'   the corresponding eigenvector is calculated.
#' @param options Options for the ARPACK solver. See
#'   [arpack_defaults()].
#' @return Depends on the algorithm used.
#'
#'   For `arpack` a list with three entries is returned:
#'   \describe{
#'     \item{options}{
#'       See the return value for `arpack()` for a complete description.
#'     }
#'     \item{values}{
#'       Numeric vector, the eigenvalues.
#'     }
#'     \item{vectors}{
#'       Numeric matrix, with the eigenvectors as columns.
#'     }
#'   }
#' @author Gabor Csardi \email{csardi.gabor@@gmail.com}
#' @seealso [as_adjacency_matrix()] to create a (sparse) adjacency matrix.
#' @keywords graphs
#' @examples
#'
#' ## Small example graph, leading eigenvector by default
#' kite <- make_graph("Krackhardt_kite")
#' spectrum(kite)[c("values", "vectors")]
#'
#' ## Double check
#' eigen(as_adjacency_matrix(kite, sparse = FALSE))$vectors[, 1]
#'
#' ## Should be the same as 'eigen_centrality' (but rescaled)
#' cor(eigen_centrality(kite)$vector, spectrum(kite)$vectors)
#'
#' ## Smallest eigenvalues
#' spectrum(kite, which = list(pos = "SM", howmany = 2))$values
#'
#' @family centrality
#' @export
spectrum <- function(
  graph,
  algorithm = c(
    "arpack",
    "auto",
    "lapack",
    "comp_auto",
    "comp_lapack",
    "comp_arpack"
  ),
  which = list(),
  options = arpack_defaults()
) {
  if (is.function(options)) {
    lifecycle::deprecate_soft(
      "1.6.0",
      "spectrum(options = 'must be a list')",
      details = c(
        "`arpack_defaults()` is now a function, use `options = arpack_defaults()` instead of `options = arpack_defaults`."
      )
    )
    options <- options()
  }

  eigen_adjacency_impl(
    graph,
    algorithm = algorithm,
    which = which,
    options = options
  )
}

eigen_defaults <- function() {
  list(
    pos = "LM",
    howmany = 1L,
    il = -1L,
    iu = -1L,
    vl = -Inf,
    vu = Inf,
    vestimate = 0L,
    balance = "none"
  )
}

#' Eigenvector centrality of vertices
#'
#' `eigen_centrality()` takes a graph (`graph`) and returns the
#' eigenvector centralities of the vertices `v` within it.
#'
#' Eigenvector centrality scores correspond to the values of the principal
#' eigenvector of the graph's adjacency matrix; these scores may, in turn, be
#' interpreted as arising from a reciprocal process in which the centrality of
#' each actor is proportional to the sum of the centralities of those actors to
#' whom he or she is connected.  In general, vertices with high eigenvector
#' centralities are those which are connected to many other vertices which are,
#' in turn, connected to many others (and so on).  The perceptive may realize
#' that this implies that the largest values will be obtained by individuals in
#' large cliques (or high-density substructures).  This is also intelligible
#' from an algebraic point of view, with the first eigenvector being closely
#' related to the best rank-1 approximation of the adjacency matrix (a
#' relationship which is easy to see in the special case of a diagonalizable
#' symmetric real matrix via the \eqn{SLS^-1}{$S \Lambda S^{-1}$}
#' decomposition).
#'
#' The adjacency matrix used in the eigenvector centrality calculation assumes
#' that loop edges are counted *twice* in undirected graphs; this is because
#' each loop edge has *two* endpoints that are both connected to the same vertex,
#' and you could traverse the loop edge via either endpoint.
#'
#' In the directed case, the left eigenvector of the adjacency matrix is
#' calculated. In other words, the centrality of a vertex is proportional to
#' the sum of centralities of vertices pointing to it.
#'
#' Eigenvector centrality is meaningful only for (strongly) connected graphs.
#' Undirected graphs that are not connected should be decomposed into connected
#' components, and the eigenvector centrality calculated for each separately.
#' This function does not verify that the graph is connected. If it is not, in
#' the undirected case the scores of all but one component will be zeros.
#'
#' Also note that the adjacency matrix of a directed acyclic graph or the
#' adjacency matrix of an empty graph does not possess positive eigenvalues,
#' therefore the eigenvector centrality is not defined for these graphs.
#' igraph will return an eigenvalue of zero in such cases. The eigenvector
#' centralities will all be equal for an empty graph and will all be zeros for
#' a directed acyclic graph. Such pathological cases can be detected by checking
#' whether the eigenvalue is very close to zero.
#'
#' From igraph version 0.5 this function uses ARPACK for the underlying
#' computation, see [arpack()] for more about ARPACK in igraph.
#'
#' @param graph Graph to be analyzed.
#' @param directed Logical scalar, whether to consider direction of the edges
#'   in directed graphs. It is ignored for undirected graphs.
#' @param scale `r lifecycle::badge("deprecated")` Normalization will always take
#' place.
#' @param weights A numerical vector or `NULL`. This argument can be used
#'   to give edge weights for calculating the weighted eigenvector centrality of
#'   vertices. If this is `NULL` and the graph has a `weight` edge
#'   attribute then that is used. If `weights` is a numerical vector then it is
#'   used, even if the graph has a `weight` edge attribute. If this is
#'   `NA`, then no edge weights are used (even if the graph has a
#'   `weight` edge attribute). Note that if there are negative edge weights
#'   and the direction of the edges is considered, then the eigenvector might be
#'   complex. In this case only the real part is reported.
#'   This function interprets weights as connection strength. Higher
#'   weights spread the centrality better.
#' @param options A named list, to override some ARPACK options. See
#'   [arpack()] for details.
#' @return A named list with components:
#'   \describe{
#'     \item{vector}{
#'       A vector containing the centrality scores.
#'     }
#'     \item{value}{
#'       The eigenvalue corresponding to the calculated eigenvector, i.e. the centrality scores.
#'     }
#'     \item{options}{
#'       A named list, information about the underlying ARPACK computation. See [arpack()] for the details.
#'     }
#'   }
#' @author Gabor Csardi \email{csardi.gabor@@gmail.com} and Carter T. Butts
#' (<http://www.faculty.uci.edu/profile.cfm?faculty_id=5057>) for the
#' manual page.
#' @references Bonacich, P.  (1987).  Power and Centrality: A Family of
#' Measures. *American Journal of Sociology*, 92, 1170-1182.
#' @keywords graphs
#' @examples
#'
#' # Generate some test data
#' g <- make_ring(10, directed = FALSE)
#' # Compute eigenvector centrality scores
#' eigen_centrality(g)
#' @family centrality
#' @export
#' @cdocs igraph_eigenvector_centrality
eigen_centrality <- function(
  graph,
  directed = FALSE,
  scale = deprecated(),
  weights = NULL,
  options = arpack_defaults()
) {
  if (is.function(options)) {
    lifecycle::deprecate_soft(
      "1.6.0",
      "eigen_centrality(options = 'must be a list')",
      details = c(
        "`arpack_defaults()` is now a function, use `options = arpack_defaults()` instead of `options = arpack_defaults`."
      )
    )
    options <- options()
  }

  if (lifecycle::is_present(scale)) {
    if (isTRUE(scale)) {
      lifecycle::deprecate_soft(
        "2.1.1",
        "eigen_centrality(scale)",
        details = "eigen_centrality() will always behave as if scale=TRUE were used."
      )
    } else {
      lifecycle::deprecate_warn(
        "2.1.1",
        "eigen_centrality(scale = 'always as if TRUE')",
        details = "Normalization is always performed"
      )
    }
  }

  eigenvector_centrality_impl(
    graph = graph,
    directed = directed,
    scale = TRUE,
    weights = weights,
    options = options
  )
}

#' Strength or weighted vertex degree
#'
#' Summing up the edge weights of the adjacent edges for each vertex.
#'
#'
#' @param graph The input graph.
#' @param vids The vertices for which the strength will be calculated.
#' @param mode Character string, \dQuote{out} for out-degree, \dQuote{in} for
#'   in-degree or \dQuote{all} for the sum of the two. For undirected graphs this
#'   argument is ignored.
#' @param loops Logical; whether the loop edges are also counted.
#' @param weights Weight vector. If the graph has a `weight` edge
#'   attribute, then this is used by default. If the graph does not have a
#'   `weight` edge attribute and this argument is `NULL`, then a
#'   [degree()] is called. If this is `NA`, then no edge weights are used
#'   (even if the graph has a `weight` edge attribute).
#' @return A numeric vector giving the strength of the vertices.
#' @author Gabor Csardi \email{csardi.gabor@@gmail.com}
#' @seealso [degree()] for the unweighted version.
#' @references Alain Barrat, Marc Barthelemy, Romualdo Pastor-Satorras,
#' Alessandro Vespignani: The architecture of complex weighted networks, Proc.
#' Natl. Acad. Sci. USA 101, 3747 (2004)
#' @keywords graphs
#' @examples
#'
#' g <- make_star(10)
#' E(g)$weight <- seq(ecount(g))
#' strength(g)
#' strength(g, mode = "out")
#' strength(g, mode = "in")
#'
#' # No weights
#' g <- make_ring(10)
#' strength(g)
#' @family centrality
#' @export
#' @cdocs igraph_strength
strength <- strength_impl


#' Graph diversity
#'
#' Calculates a measure of diversity for all vertices.
#'
#' The diversity of a vertex is defined as the (scaled) Shannon entropy of the
#' weights of its incident edges:
#' \deqn{D(i)=\frac{H(i)}{\log k_i}}{D(i)=H(i)/log(k[i])}
#' and
#' \deqn{H(i)=-\sum_{j=1}^{k_i} p_{ij}\log p_{ij},}{H(i) =
#'   -sum(p[i,j] log(p[i,j]), j=1..k[i]),} where
#' \deqn{p_{ij}=\frac{w_{ij}}{\sum_{l=1}^{k_i}}V_{il},}{p[i,j] = w[i,j] /
#' sum(w[i,l], l=1..k[i]),} and \eqn{k_i}{k[i]} is the (total) degree of vertex
#' \eqn{i}, \eqn{w_{ij}}{w[i,j]} is the weight of the edge(s) between vertices
#' \eqn{i} and \eqn{j}.
#'
#' For vertices with degree less than two the function returns `NaN`.
#'
#' @param graph The input graph. Edge directions are ignored.
#' @param weights `NULL`, or the vector of edge weights to use for the
#'   computation. If `NULL`, then the \sQuote{weight} attibute is used. Note
#'   that this measure is not defined for unweighted graphs.
#' @param vids The vertex ids for which to calculate the measure.
#' @return A numeric vector, its length is the number of vertices.
#' @author Gabor Csardi \email{csardi.gabor@@gmail.com}
#' @references Nathan Eagle, Michael Macy and Rob Claxton: Network Diversity
#' and Economic Development, *Science* **328**, 1029--1031, 2010.
#' @keywords graphs
#' @examples
#'
#' g1 <- sample_gnp(20, 2 / 20)
#' g2 <- sample_gnp(20, 2 / 20)
#' g3 <- sample_gnp(20, 5 / 20)
#' E(g1)$weight <- 1
#' E(g2)$weight <- runif(ecount(g2))
#' E(g3)$weight <- runif(ecount(g3))
#' diversity(g1)
#' diversity(g2)
#' diversity(g3)
#' @family centrality
#' @export
#' @cdocs igraph_diversity
diversity <- diversity_impl


#' Kleinberg's hub and authority centrality scores.
#'
#' The hub scores of the vertices are defined as the principal eigenvector
#' of \eqn{A A^T}{A*t(A)}, where \eqn{A} is the adjacency matrix of the
#' graph.
#'
#' Similarly, the authority scores of the vertices are defined as the principal
#' eigenvector of \eqn{A^T A}{t(A)*A}, where \eqn{A} is the adjacency matrix of
#' the graph.
#'
#' For undirected matrices the adjacency matrix is symmetric and the hub
#' scores are the same as authority scores.
#'
#' @param graph The input graph.
#' @param scale Logical scalar, whether to scale the result to have a maximum
#'   score of one. If no scaling is used then the result vector has unit length
#'   in the Euclidean norm.
#' @param weights Optional positive weight vector for calculating weighted
#'   scores. If the graph has a `weight` edge attribute, then this is used
#'   by default. Pass `NA` to ignore the weight attribute. This function
#'   interprets edge weights as connection strengths. The weights of parallel
#'   edges are effectively added up.
#' @param options A named list, to override some ARPACK options. See
#'   [arpack()] for details.
#' @inheritParams rlang::args_dots_empty
#' @return A named list with members:
#'   \describe{
#'     \item{hub}{
#'       The hub score of the vertices.
#'     }
#'     \item{authority}{
#'       The authority score of the vertices.
#'     }
#'     \item{value}{
#'       The corresponding eigenvalue of the calculated principal eigenvector.
#'     }
#'     \item{options}{
#'       Some information about the ARPACK computation, it has the same members as the `options` member returned by [arpack()], see that for documentation.
#'     }
#'   }
#' @seealso [eigen_centrality()] for eigenvector centrality,
#' [page_rank()] for the Page Rank scores. [arpack()] for
#' the underlining machinery of the computation.
#' @references J. Kleinberg. Authoritative sources in a hyperlinked
#' environment. *Proc. 9th ACM-SIAM Symposium on Discrete Algorithms*,
#' 1998. Extended version in *Journal of the ACM* 46(1999). Also appears
#' as IBM Research Report RJ 10076, May 1997.
#'
#' @export
#' @examples
#' ## An in-star
#' g <- make_star(10)
#' hits_scores(g)
#'
#' ## A ring
#' g2 <- make_ring(10)
#' hits_scores(g2)
#' @family centrality
#' @cdocs igraph_hub_and_authority_scores
hits_scores <- function(
  graph,
  ...,
  scale = TRUE,
  weights = NULL,
  options = arpack_defaults()
) {
  rlang::check_dots_empty()

  hub_and_authority_scores_impl(
    graph = graph,
    scale = scale,
    weights = weights,
    options = options
  )
}

#' @title Kleinberg's authority centrality scores.
#' @rdname hub_score
#' @param options A named list, to override some ARPACK options. See
#'   [arpack()] for details.
#' @export
authority_score <- function(
  graph,
  scale = TRUE,
  weights = NULL,
  options = arpack_defaults()
) {
  lifecycle::deprecate_soft("2.1.0", "authority_score()", "hits_scores()")
  if (is.function(options)) {
    lifecycle::deprecate_soft(
      "1.6.0",
      I("arpack_defaults"),
      "arpack_defaults()",
      details = c(
        "So the function arpack_defaults(), not an object called arpack_defaults."
      )
    )
    options <- arpack_defaults()
  }

  scores <- hits_scores(
    graph = graph,
    scale = scale,
    weights = weights,
    options = options
  )
  scores$hub <- NULL
  rlang::set_names(scores, c("vector", "value", "options"))
}

#' @title Kleinberg's hub centrality scores.
#' @rdname hub_score
#' @param graph The input graph.
#' @param scale Logical scalar, whether to scale the result to have a maximum
#'   score of one. If no scaling is used then the result vector has unit length
#'   in the Euclidean norm.
#' @param weights Optional positive weight vector for calculating weighted
#'   scores. If the graph has a `weight` edge attribute, then this is used
#'   by default.
#'   This function interprets edge weights as connection strengths. In the
#'   random surfer model, an edge with a larger weight is more likely to be
#'   selected by the surfer.
#' @param options A named list, to override some ARPACK options. See
#'   [arpack()] for details.
#' @family centrality
#' @export
hub_score <- function(
  graph,
  scale = TRUE,
  weights = NULL,
  options = arpack_defaults()
) {
  lifecycle::deprecate_soft("2.0.3", "hub_score()", "hits_scores()")
  if (is.function(options)) {
    lifecycle::deprecate_soft(
      "1.6.0",
      I("arpack_defaults"),
      "arpack_defaults()",
      details = c(
        "So the function arpack_defaults(), not an object called arpack_defaults."
      )
    )
    options <- arpack_defaults()
  }

  scores <- hits_scores(
    graph = graph,
    scale = scale,
    weights = weights,
    options = options
  )
  scores$authority <- NULL
  rlang::set_names(scores, c("vector", "value", "options"))
}

#' The Page Rank algorithm
#'
#' Calculates the Google PageRank for the specified vertices.
#'
#' For the explanation of the PageRank algorithm, see the following webpage:
#' <http://infolab.stanford.edu/~backrub/google.html>, or the following
#' reference:
#'
#' Sergey Brin and Larry Page: The Anatomy of a Large-Scale Hypertextual Web
#' Search Engine. Proceedings of the 7th World-Wide Web Conference, Brisbane,
#' Australia, April 1998.
#'
#' The `page_rank()` function can use either the PRPACK library or ARPACK
#' (see [arpack()]) to perform the calculation.
#'
#' Please note that the PageRank of a given vertex depends on the PageRank of
#' all other vertices, so even if you want to calculate the PageRank for only
#' some of the vertices, all of them must be calculated. Requesting the
#' PageRank for only some of the vertices does not result in any performance
#' increase at all.
#'
#' @param graph The graph object.
#' @param algo Character scalar, which implementation to use to carry out the
#'   calculation. The default is `"prpack"`, which uses the PRPACK library
#'   (<https://github.com/dgleich/prpack>) to calculate PageRank scores
#'   by solving a set of linear equations. This is a new implementation in igraph
#'   version 0.7, and the suggested one, as it is the most stable and the fastest
#'   for all but small graphs.  `"arpack"` uses the ARPACK library, the
#'   default implementation from igraph version 0.5 until version 0.7. It computes
#'   PageRank scores by solving an eingevalue problem.
#' @param vids The vertices of interest.
#' @param directed Logical, if true directed paths will be considered for
#'   directed graphs. It is ignored for undirected graphs.
#' @param damping The damping factor (\sQuote{d} in the original paper).
#' @param personalized Optional vector giving a probability distribution to
#'   calculate personalized PageRank. For personalized PageRank, the probability
#'   of jumping to a node when abandoning the random walk is not uniform, but it
#'   is given by this vector. The vector should contains an entry for each vertex
#'   and it will be rescaled to sum up to one.
#' @param weights A numerical vector or `NULL`. This argument can be used
#'   to give edge weights for calculating the weighted PageRank of vertices. If
#'   this is `NULL` and the graph has a `weight` edge attribute then
#'   that is used. If `weights` is a numerical vector then it used, even if
#'   the graph has a `weights` edge attribute. If this is `NA`, then no
#'   edge weights are used (even if the graph has a `weight` edge attribute.
#'   This function interprets edge weights as connection strengths. In the
#'   random surfer model, an edge with a larger weight is more likely to be
#'   selected by the surfer.
#' @param options A named list, to override some ARPACK options. See
#'   [arpack()] for details. This argument is ignored if the PRPACK
#'   implementation is used.
#' @return A named list with entries:
#'   \describe{
#'     \item{vector}{
#'       A numeric vector with the PageRank scores.
#'     }
#'     \item{value}{
#'       When using the ARPACK method, the eigenvalue corresponding to the eigenvector with the PageRank scores is returned here. It is expected to be exactly one, and can be used to check that ARPACK has successfully converged to the expected eingevector. When using the PRPACK method, it is always set to 1.0.
#'     }
#'     \item{options}{
#'       Some information about the underlying ARPACK calculation. See [arpack()] for details. This entry is `NULL` if not the ARPACK implementation was used.
#'     }
#'   }
#'
#' @author Tamas Nepusz \email{ntamas@@gmail.com} and Gabor Csardi
#' \email{csardi.gabor@@gmail.com}
#' @seealso Other centrality scores: [closeness()],
#' [betweenness()], [degree()]
#' @references Sergey Brin and Larry Page: The Anatomy of a Large-Scale
#' Hypertextual Web Search Engine. Proceedings of the 7th World-Wide Web
#' Conference, Brisbane, Australia, April 1998.
#' @keywords graphs
#' @examples
#'
#' g <- sample_gnp(20, 5 / 20, directed = TRUE)
#' page_rank(g)$vector
#'
#' g2 <- make_star(10)
#' page_rank(g2)$vector
#'
#' # Personalized PageRank
#' g3 <- make_ring(10)
#' page_rank(g3)$vector
#' reset <- seq(vcount(g3))
#' page_rank(g3, personalized = reset)$vector
#' @family centrality
#' @export
#' @cdocs igraph_personalized_pagerank
page_rank <- personalized_pagerank_impl

#' Harmonic centrality of vertices
#'
#' The harmonic centrality of a vertex is the mean inverse distance to all other
#' vertices. The inverse distance to an unreachable vertex is considered to be zero.
#'
#' The `cutoff` argument can be used to restrict the calculation to paths
#' of length `cutoff` or smaller only; this can be used for larger graphs
#' to speed up the calculation. If `cutoff` is negative (which is the
#' default), then the function calculates the exact harmonic centrality scores.
#'
#' @param graph The graph to analyze.
#' @param vids The vertices for which harmonic centrality will be calculated.
#' @param mode Character string, defining the types of the paths used for
#'   measuring the distance in directed graphs. \dQuote{out} follows paths along
#'   the edge directions only, \dQuote{in} traverses the edges in reverse, while
#'   \dQuote{all} ignores edge directions. This argument is ignored for undirected
#'   graphs.
#' @param normalized Logical scalar, whether to calculate the normalized
#'   harmonic centrality. If true, the result is the mean inverse path length to
#'   other vertices, i.e. it is normalized by the number of vertices minus one.
#'   If false, the result is the sum of inverse path lengths to other vertices.
#' @param weights Optional positive weight vector for calculating weighted
#'   harmonic centrality. If the graph has a `weight` edge attribute, then
#'   this is used by default. Weights are used for calculating weighted shortest
#'   paths, so they are interpreted as distances.
#' @param cutoff The maximum path length to consider when calculating the
#'   harmonic centrality. There is no such limit when the cutoff is negative. Note that
#'   zero cutoff means that only paths of at most length 0 are considered.
#' @return Numeric vector with the harmonic centrality scores of all the vertices in
#'   `v`.
#' @seealso [betweenness()], [closeness()]
#' @references M. Marchiori and V. Latora, Harmony in the small-world,
#' *Physica A* 285, pp. 539-546 (2000).
#' @family centrality
#' @export
#' @keywords graphs
#' @examples
#'
#' g <- make_ring(10)
#' g2 <- make_star(10)
#' harmonic_centrality(g)
#' harmonic_centrality(g2, mode = "in")
#' harmonic_centrality(g2, mode = "out")
#' harmonic_centrality(g %du% make_full_graph(5), mode = "all")
#'
#' @cdocs igraph_harmonic_centrality_cutoff
harmonic_centrality <- harmonic_centrality_cutoff_impl


bonpow.dense <- function(
  graph,
  nodes = V(graph),
  loops = FALSE,
  exponent = 1,
  rescale = FALSE,
  tol = 1e-7
) {
  ensure_igraph(graph)

  d <- as_adjacency_matrix(graph)
  if (!loops) {
    diag(d) <- 0
  }
  n <- vcount(graph)
  id <- matrix(0, nrow = n, ncol = n)
  diag(id) <- 1

  #  ev <- apply(solve(id-exponent*d,tol=tol)%*%d,1,sum)
  ev <- solve(id - exponent * d, tol = tol) %*% apply(d, 1, sum)
  if (rescale) {
    ev <- ev / sum(ev)
  } else {
    ev <- ev * sqrt(n / sum((ev)^2))
  }
  ev[as.numeric(nodes)]
}

bonpow.sparse <- function(
  graph,
  nodes = V(graph),
  loops = FALSE,
  exponent = 1,
  rescale = FALSE,
  tol = 1e-07
) {
  ## remove loops if requested
  if (!loops) {
    graph <- simplify(graph, remove.multiple = FALSE, remove.loops = TRUE)
  }

  vg <- vcount(graph)

  ## sparse adjacency matrix
  d <- as_adjacency_matrix(graph, sparse = TRUE)

  ## sparse identity matrix
  id <- as(Matrix::Matrix(diag(vg), doDiag = FALSE), "generalMatrix")

  ## solve it
  ev <- Matrix::solve(id - exponent * d, degree(graph, mode = "out"), tol = tol)

  if (rescale) {
    ev <- ev / sum(ev)
  } else {
    ev <- ev * sqrt(vcount(graph) / sum((ev)^2))
  }

  ev[as.numeric(nodes)]
}


#' Find Bonacich Power Centrality Scores of Network Positions
#'
#' `power_centrality()` takes a graph (`dat`) and returns the Boncich power
#' centralities of positions (selected by `nodes`).  The decay rate for
#' power contributions is specified by `exponent` (1 by default).
#'
#' Bonacich's power centrality measure is defined by
#' \eqn{C_{BP}\left(\alpha,\beta\right)=\alpha\left(\mathbf{I}-\beta\mathbf{A}\right)^{-1}\mathbf{A}\mathbf{1}}{C_BP(alpha,beta)=alpha
#' (I-beta A)^-1 A 1}, where \eqn{\beta}{beta} is an attenuation parameter (set
#' here by `exponent`) and \eqn{\mathbf{A}}{A} is the graph adjacency
#' matrix.  (The coefficient \eqn{\alpha}{alpha} acts as a scaling parameter,
#' and is set here (following Bonacich (1987)) such that the sum of squared
#' scores is equal to the number of vertices.  This allows 1 to be used as a
#' reference value for the ``middle'' of the centrality range.)  When
#' \eqn{\beta \rightarrow }{beta->1/lambda_A1}\eqn{
#' 1/\lambda_{\mathbf{A}1}}{beta->1/lambda_A1} (the reciprocal of the largest
#' eigenvalue of \eqn{\mathbf{A}}{A}), this is to within a constant multiple of
#' the familiar eigenvector centrality score; for other values of \eqn{\beta},
#' the behavior of the measure is quite different.  In particular, \eqn{\beta}
#' gives positive and negative weight to even and odd walks, respectively, as
#' can be seen from the series expansion
#' \eqn{C_{BP}\left(\alpha,\beta\right)=\alpha \sum_{k=0}^\infty \beta^k
#' }{C_BP(alpha,beta) = alpha sum( beta^k A^(k+1) 1, k in 0..infinity )}\eqn{
#' \mathbf{A}^{k+1} \mathbf{1}}{C_BP(alpha,beta) = alpha sum( beta^k A^(k+1) 1,
#' k in 0..infinity )} which converges so long as \eqn{|\beta|
#' }{|beta|<1/lambda_A1}\eqn{ < 1/\lambda_{\mathbf{A}1}}{|beta|<1/lambda_A1}.
#' The magnitude of \eqn{\beta}{beta} controls the influence of distant actors
#' on ego's centrality score, with larger magnitudes indicating slower rates of
#' decay.  (High rates, hence, imply a greater sensitivity to edge effects.)
#'
#' Interpretively, the Bonacich power measure corresponds to the notion that
#' the power of a vertex is recursively defined by the sum of the power of its
#' alters.  The nature of the recursion involved is then controlled by the
#' power exponent: positive values imply that vertices become more powerful as
#' their alters become more powerful (as occurs in cooperative relations),
#' while negative values imply that vertices become more powerful only as their
#' alters become *weaker* (as occurs in competitive or antagonistic
#' relations).  The magnitude of the exponent indicates the tendency of the
#' effect to decay across long walks; higher magnitudes imply slower decay.
#' One interesting feature of this measure is its relative instability to
#' changes in exponent magnitude (particularly in the negative case).  If your
#' theory motivates use of this measure, you should be very careful to choose a
#' decay parameter on a non-ad hoc basis.
#'
#' For directed networks, the Bonacich power measure can be understood as
#' similar to status in the network where higher status nodes have more edges
#' that point from them to others with status. Node A's centrality depends
#' on the centrality of all the nodes that A points toward, and their centrality
#' depends on the nodes they point toward, etc. Note, this means that a node
#' with an out-degree of 0 will have a Bonacich power centrality of 0 as they
#' do not point towards anyone. When using this with directed network it
#' is important to think about the edge direction and what it represents.
#'
#' @param graph the input graph.
#' @param nodes vertex sequence indicating which vertices are to be included in
#'   the calculation.  By default, all vertices are included.
#' @param loops boolean indicating whether or not the diagonal should be
#'   treated as valid data.  Set this true if and only if the data can contain
#'   loops.  `loops` is `FALSE` by default.
#' @param exponent exponent (decay rate) for the Bonacich power centrality
#'   score; can be negative
#' @param rescale if true, centrality scores are rescaled such that they sum to
#'   1.
#' @param tol tolerance for near-singularities during matrix inversion (see
#'   [solve()])
#' @param sparse Logical scalar, whether to use sparse matrices for the
#'   calculation. The \sQuote{Matrix} package is required for sparse matrix
#'   support
#' @return A vector, containing the centrality scores.
#' @note This function was ported (i.e. copied) from the SNA package.
#' @section Warning : Singular adjacency matrices cause no end of headaches for
#' this algorithm; thus, the routine may fail in certain cases.  This will be
#' fixed when we get a better algorithm.
#' @author Carter T. Butts
#' (<http://www.faculty.uci.edu/profile.cfm?faculty_id=5057>), ported to
#' igraph by Gabor Csardi \email{csardi.gabor@@gmail.com}
#' @seealso [eigen_centrality()] and [alpha_centrality()]
#' @references Bonacich, P.  (1972).  ``Factoring and Weighting Approaches to
#' Status Scores and Clique Identification.'' *Journal of Mathematical
#' Sociology*, 2, 113-120.
#'
#' Bonacich, P.  (1987).  ``Power and Centrality: A Family of Measures.''
#' *American Journal of Sociology*, 92, 1170-1182.
#' @keywords graphs
#' @family centrality
#' @export
#' @examples
#'
#' # Generate some test data from Bonacich, 1987:
#' g.c <- make_graph(c(1, 2, 1, 3, 2, 4, 3, 5), dir = FALSE)
#' g.d <- make_graph(c(1, 2, 1, 3, 1, 4, 2, 5, 3, 6, 4, 7), dir = FALSE)
#' g.e <- make_graph(c(1, 2, 1, 3, 1, 4, 2, 5, 2, 6, 3, 7, 3, 8, 4, 9, 4, 10), dir = FALSE)
#' g.f <- make_graph(
#'   c(1, 2, 1, 3, 1, 4, 2, 5, 2, 6, 2, 7, 3, 8, 3, 9, 3, 10, 4, 11, 4, 12, 4, 13),
#'   dir = FALSE
#' )
#' # Compute power centrality scores
#' for (e in seq(-0.5, .5, by = 0.1)) {
#'   print(round(power_centrality(g.c, exp = e)[c(1, 2, 4)], 2))
#' }
#'
#' for (e in seq(-0.4, .4, by = 0.1)) {
#'   print(round(power_centrality(g.d, exp = e)[c(1, 2, 5)], 2))
#' }
#'
#' for (e in seq(-0.4, .4, by = 0.1)) {
#'   print(round(power_centrality(g.e, exp = e)[c(1, 2, 5)], 2))
#' }
#'
#' for (e in seq(-0.4, .4, by = 0.1)) {
#'   print(round(power_centrality(g.f, exp = e)[c(1, 2, 5)], 2))
#' }
#'
power_centrality <- function(
  graph,
  nodes = V(graph),
  loops = FALSE,
  exponent = 1,
  rescale = FALSE,
  tol = 1e-7,
  sparse = TRUE
) {
  nodes <- as_igraph_vs(graph, nodes)
  if (sparse) {
    res <- bonpow.sparse(graph, nodes, loops, exponent, rescale, tol)
  } else {
    res <- bonpow.dense(graph, nodes, loops, exponent, rescale, tol)
  }

  if (igraph_opt("add.vertex.names") && is_named(graph)) {
    names(res) <- vertex_attr(graph, "name", nodes)
  }

  res
}

alpha.centrality.dense <- function(
  graph,
  nodes = V(graph),
  alpha = 1,
  loops = FALSE,
  exo = 1,
  weights = NULL,
  tol = 1e-7
) {
  ensure_igraph(graph)

  exo <- rep(exo, length.out = vcount(graph))
  exo <- matrix(exo, ncol = 1)

  if (is.null(weights) && "weight" %in% edge_attr_names(graph)) {
    ## weights == NULL and there is a "weight" edge attribute
    attr <- "weight"
  } else if (is.null(weights)) {
    ## weights == NULL, but there is no "weight" edge attribute
    attr <- NULL
  } else if (is.character(weights) && length(weights) == 1) {
    ## name of an edge attribute, nothing to do
    attr <- "weight"
  } else if (any(!is.na(weights))) {
    ## weights != NULL and weights != rep(NA, x)
    graph <- set_edge_attr(graph, "weight", value = as.numeric(weights))
    attr <- "weight"
  } else {
    ## weights != NULL, but weights == rep(NA, x)
    attr <- NULL
  }

  d <- t(as_adjacency_matrix(graph, attr = attr, sparse = FALSE))
  if (!loops) {
    diag(d) <- 0
  }
  n <- vcount(graph)
  id <- matrix(0, nrow = n, ncol = n)
  diag(id) <- 1

  ev <- solve(id - alpha * d, tol = tol) %*% exo
  ev[as.numeric(nodes)]
}

alpha.centrality.sparse <- function(
  graph,
  nodes = V(graph),
  alpha = 1,
  loops = FALSE,
  exo = 1,
  weights = NULL,
  tol = 1e-7
) {
  ensure_igraph(graph)

  vc <- vcount(graph)

  if (!loops) {
    graph <- simplify(graph, remove.multiple = FALSE, remove.loops = TRUE)
  }

  if (is.null(weights) && "weight" %in% edge_attr_names(graph)) {
    ## weights == NULL and there is a "weight" edge attribute
    attr <- "weight"
  } else if (is.null(weights)) {
    ## weights == NULL, but there is no "weight" edge attribute
    attr <- NULL
  } else if (is.character(weights) && length(weights) == 1) {
    ## name of an edge attribute, nothing to do
    attr <- "weight"
  } else if (any(!is.na(weights))) {
    ## weights != NULL and weights != rep(NA, x)
    graph <- set_edge_attr(graph, "weight", value = as.numeric(weights))
    attr <- "weight"
  } else {
    ## weights != NULL, but weights == rep(NA, x)
    attr <- NULL
  }

  M <- Matrix::t(as_adjacency_matrix(graph, attr = attr, sparse = TRUE))

  ## Create an identity matrix
  M2 <- Matrix::sparseMatrix(
    dims = c(vc, vc),
    i = 1:vc,
    j = 1:vc,
    x = rep(1, vc)
  )

  ## exo
  exo <- cbind(rep(exo, length.out = vc))

  ## Solve the equation
  M3 <- M2 - alpha * M
  r <- Matrix::solve(M3, tol = tol, exo)

  r[as.numeric(nodes)]
}


#' Find Bonacich alpha centrality scores of network positions
#'
#' `alpha_centrality()` calculates the alpha centrality of some (or all)
#' vertices in a graph.
#'
#' The alpha centrality measure can be considered as a generalization of
#' eigenvector centrality to directed graphs. It was proposed by Bonacich in
#' 2001 (see reference below).
#'
#' The alpha centrality of the vertices in a graph is defined as the solution
#' of the following matrix equation: \deqn{x=\alpha A^T x+e,}{x=alpha t(A)x+e,}
#' where \eqn{A}{A} is the (not necessarily symmetric) adjacency matrix of the
#' graph, \eqn{e}{e} is the vector of exogenous sources of status of the
#' vertices and \eqn{\alpha}{alpha} is the relative importance of the
#' endogenous versus exogenous factors.
#'
#' @param graph The input graph, can be directed or undirected. In undirected
#'   graphs, edges are treated as if they were reciprocal directed ones.
#' @param nodes Vertex sequence, the vertices for which the alpha centrality
#'   values are returned. (For technical reasons they will be calculated for all
#'   vertices, anyway.)
#' @param alpha Parameter specifying the relative importance of endogenous
#'   versus exogenous factors in the determination of centrality. See details
#'   below.
#' @param loops Whether to eliminate loop edges from the graph before the
#'   calculation.
#' @param exo The exogenous factors, in most cases this is either a constant --
#'   the same factor for every node, or a vector giving the factor for every
#'   vertex. Note that too long vectors will be truncated and too short vectors
#'   will be replicated to match the number of vertices.
#' @param weights A character scalar that gives the name of the edge attribute
#'   to use in the adjacency matrix. If it is `NULL`, then the
#'   \sQuote{weight} edge attribute of the graph is used, if there is one.
#'   Otherwise, or if it is `NA`, then the calculation uses the standard
#'   adjacency matrix.
#' @param tol Tolerance for near-singularities during matrix inversion, see
#'   [solve()].
#' @param sparse Logical scalar, whether to use sparse matrices for the
#'   calculation. The \sQuote{Matrix} package is required for sparse matrix
#'   support
#' @return A numeric vector contaning the centrality scores for the selected
#'   vertices.
#' @section Warning: Singular adjacency matrices cause problems for this
#' algorithm, the routine may fail is certain cases.
#' @author Gabor Csardi \email{csardi.gabor@@gmail.com}
#' @seealso [eigen_centrality()] and [power_centrality()]
#' @references Bonacich, P. and Lloyd, P. (2001). ``Eigenvector-like
#' measures of centrality for asymmetric relations'' *Social Networks*,
#' 23, 191-201.
#' @family centrality
#' @export
#' @keywords graphs
#' @examples
#'
#' # The examples from Bonacich's paper
#' g.1 <- make_graph(c(1, 3, 2, 3, 3, 4, 4, 5))
#' g.2 <- make_graph(c(2, 1, 3, 1, 4, 1, 5, 1))
#' g.3 <- make_graph(c(1, 2, 2, 3, 3, 4, 4, 1, 5, 1))
#' alpha_centrality(g.1)
#' alpha_centrality(g.2)
#' alpha_centrality(g.3, alpha = 0.5)
#'
alpha_centrality <- function(
  graph,
  nodes = V(graph),
  alpha = 1,
  loops = FALSE,
  exo = 1,
  weights = NULL,
  tol = 1e-7,
  sparse = TRUE
) {
  nodes <- as_igraph_vs(graph, nodes)
  if (sparse) {
    res <- alpha.centrality.sparse(
      graph,
      nodes,
      alpha,
      loops,
      exo,
      weights,
      tol
    )
  } else {
    res <- alpha.centrality.dense(
      graph,
      nodes,
      alpha,
      loops,
      exo,
      weights,
      tol
    )
  }
  if (igraph_opt("add.vertex.names") && is_named(graph)) {
    names(res) <- vertex_attr(graph, "name", nodes)
  }
  res
}